Single screw extrusion of viscoplastic fluids subject to different slip coefficients at screw and barrel surfaces

Commonly encountered viscoplastic fluids including concentrated suspensions of polymeric and ceramic composites, foams, gels, concrete, food products, and energetic compounds exhibit wall slip during their flow and processing. For some viscoplastics fluids, especially highly filled suspensions, wall slip may dominate the flow and deformation and hence the processing behavior of the suspension. The wall slip velocity is generally a function of the wall shear stress and temperature. Various factors including the materials of construction i.e., chemical nature and the roughness of the wall surface affect the wall slip behavior of viscoplastic fluids. In this study an analytical model of the extrusion of viscoplastic fluids under isothermal and fully-developed conditions in shallow channels is developed. The model accommodates the use of different slip coefficients at barrel and screw surfaces. It thus permits the investigation of effects of introducing different materials of construction for the barrel and screw surfaces and development of design expressions.     

EXTRUSION AND LUBRICATION FLOWS OF VISCOPLASTIC FLUIDS WITH WALL SLIP

The simplest model flow which approximates the extrusion (shallow screw channels) and lubrication flow is the steady, laminar flow occurring between two infinitely long parallel plates i.e., the generalized plane Couette flow. Here we develop an analytical model of the generalized plane Couette flow of viscoplastic fluids. The deformation and flow behavior of viscoplastic fluids can be realistically represented with the Herschel-Bulkley constitutive equation, which we have utilized as the basis for the development of our analytical model. Furthermore, as also demonstrated here, the deformation behavior of viscoplastic fluids is generally complicated by the presence of wall slip at solid walls, which occurs as a function of the wall shear stress. The wall slip versus the wall shear stress behavior of viscoplastic fluids can be experimentally characterized using viscomelric flows, including steady torsional and capillary flows. Thus determined Navier's wall slip coefficient can then be utilized in modeling of processing flows. In our analytical model of the generalized plane Couette flow of viscoplastic fluids the Navier's wall slip boundary condition was included. This model should be an important engineering tool, which provides design expressions for the extrusion and lubrication flows of viscoplastic fluids, with or without wall slip occurring at the walls. @KEYWORDS:Extrusion, lubrication, flow, viscoplastic, slip.     

Squeezing flow of viscoplastic fluids subject to wall slip

Polymer processing operations such as compression molding, sheet forming and injection molding can be modeled by squeezing flows between two approaching parallel surfaces in relative motion. Squeezing flows also find applications in the modeling of lubrication systems, and in the determination of rheological properties. Here, analytical solutions are developed for the constant-speed squeezing flow of viscoplastic fluids. It is assumed that the fluid is purely viscous, and hence viscoelastic effects unimportant. The rheological behavior of the viscoplastic fluids is represented by the Herschel-Bulkley viscosity function. The deformation behavior of commonly encountered viscoplastic fluids is generally complicated by the presence of wall slip at solid walls, which is a function of the wall shear stress. The slip coefficient that relates the slip velocity to the shear stress is affected by the material of construction and also the roughness of the solid surfaces, leading to the possibility of different slip coefficients at various solid surfaces. The model developed in this study accommodates the use of different slip coefficients at different solid surfaces. The accuracy of the solutions is established, and the effects of various parameters such as slip coefficient and apparent yield stress are examined. The solutions provide useful design expressions that can be utilized for squeezing flows of viscoplastic fluids, with or without wall slip at the solid boundaries.     

A dual slit in‐line die for measuring the flow properties of S‐PVC formulations

We have developed an instrumented dual slit die mounted on a twin‐screw extruder. This device allows us to distinguish the predominant flow pattern and calculate the shear viscosity, Cogswell elongational viscosity, and a Mooney wall‐slip velocity. The melt‐down process is also monitored by measuring the screw torque together with temperatures and pressures along the screw barrel. So far, we have seen that many pipe and profile formulations have a predominant plug or slip‐dominated flow behavior in the die, while others can be more sticky. Generally, the sticky highly viscous formulations will be more affected by shear heating effects when exposed to high rates during processing. We also give a detailed discussion, with examples, of how data from the device are to be analyzed and how the correct flow boundary condition is to be identified.     

Sharkskin and oscillating melt fracture: Why in slit and capillary dies and not in annular dies?

The sharkskin and stick‐slip polymer extrusion instabilities are studied primarily as functions of the type of die geometry. Experimental observations concerning the flow curves, the critical wall shear stress for the onset of the instabilities, the pressure and flow rate oscillations, and the effects of geometry and operating conditions are presented for linear low‐density polyethylenes. It is found that sharkskin and stick‐slip instabilities are present in the capillary and slit extrusion. However, annular extrusion stick‐slip and sharkskin are absent at high ratios of the inside‐to‐outside diameter of the annular die. This observation also explains the absence of these phenomena in other polymer processing operations such as film blowing. These phenomena are explained in terms of the surface‐to‐volume ratio of the extrudates, that is, if this ratio is high, sharkskin and stick‐slip are absent. POLYM. ENG. SCI., 2008. © 2007 Society of Plastics Engineers     

Influence of Rheological Behavior of Purely Viscous Fluids on Analytical Residence Time Distribution in Straight Tubes

In an effort to better understand the homogeneity of heat treatment of foodstuffs in holding tubes, the cumulative residence time distribution function is derived for a Herschel‐Bulkley fluid from fully developed laminar flow in a straight circular tube under isothermal conditions when diffusional effects are negligible. The proposed analytical solution can be reduced to solutions for Newtonian, shear‐thinning, dilatant, Bingham fluids by setting particular rheological parameters, and consequently, it is possible to successfully explain the dependence of residence time distribution on fluid properties for almost all of the rheological models used for time‐independent purely viscous fluids.     

What is the role of “pressure” in the use of capillary and slit flows to determine the shear‐rate dependent viscosity of a viscoelastic fluid?

The purpose of this review article is to clear the confusion created by some investigators, who erroneously thought that the pressure transducers mounted on the wall of a capillary or slit die measured a quantity that could meaningfully be called “pressure,” accurately stated “indeterminate isotropic contribution to the total stress,” and then reported on the effect of “pressure” on the shear‐rate dependent viscosity of a viscoelastic fluid. On the other hand, reference to such a quantity is not needed to calculate the wall shear stress and thus shear viscosity in fully developed flow of incompressible, viscoelastic polymer melts in a capillary or slit die; instead only information on the gradient of the total wall normal stress is needed. Further, it is pointed out that much of the literature discussing “pressure shift factor” to describe the effect of “pressure” on the viscosity of polymer melts in flow through a capillary or slit die is based on an erroneous belief that there exists a physically meaningful isotropic “pressure” that can be measured. POLYM. ENG. SCI., 2008. © 2008 Society of Plastics Engineers.     

Experimental study and modeling of wall slip of polymethylmethacrylate considering different die surfaces

Wall slip of polymethylmethacrylate (PMMA) was studied on different flow channel surfaces using a rheological slit die and a high pressure capillary rheometer. As die surfaces polished steel, ground steel, and Si doped Diamond like carbon (DLC) were used. A new wall slip model is presented in this paper which assumes a lubricating film between the polymer melt and the die surface. The slip velocity has a power law dependency on wall shear stress. In the double logarithmic plot the wall slip curves are linear and can be parallel shifted to higher values with increasing temperature. The predicted dependencies of the wall slip velocity could be confirmed with experiments conducted with PMMA on polished steel. Furthermore, the die surface influences the flow behavior of PMMA. No wall slip was found on ground steel and on DLC. No complete film could be established by the lubricant on the ground steel die wall. The DLC‐coating exhibits a similar surface roughness and surface energy to polished steel, but the chemical composition is different. It is a metastable form of amorphous carbon containing sp² and sp³ bonds. As a consequence slip additives have a low ability to bond to this material. POLYM. ENG. SCI., 58:1391–1398, 2018. © 2017 Society of Plastics Engineers     

Effect of wall slip on the uniformity of flow from sheeting extrusion dies

A theoretical study for analyzing the uniformity of flow from sheeting extrusion dies is presented. In this study it is assume that a slip condition exists at the wall of the die, the magnitude of slip velocity is proportional to the shear stress at the wall, the flow is isothermal and steady state, and a power law model is valid for viscosity. Two extrusion dies, T-dies and coat-hanger dies, are examined. The flow uniformity at the exit of the die is calculated and compared with that for a nonslip analysis. The discrepancies between the slip and nonslip models imply that the wall slip condition induces a significant nonuniform flow distribution. Traditional design criticism based on the nonslip model are invalid for flow with the wall slip condition, and it is necessary to increase the length of the die land to even the flow distribution at the exit of the die.     

Slip effects in tapered dies

Approximate analytical equations are derived for the calculation of pressure drop of power‐law fluids for viscous flow through tapered dies for a wide range of wall‐slip conditions. The predicted pressure drop values are compared with two‐dimensional (2D) finite element calculations to identify contraction angles for which the analytical equations can be used. It is found that the disagreement increases with increase of the contraction angle and with increase of wall slip. At a given flow rate, the pressure drop from the analytical equations is found to decrease continuously with contraction angle, which agrees with the 2D calculations only at small contraction angles. At larger contraction angles, the 2D calculations show that pressure drop increases with contraction angle as opposed to the no‐slip case where pressure drop saturates. The existence of a minimum pressure at a specific taper angle depends on the rheological parameters of the fluid and the degree of slip (slip‐law exponent), and has scientific importance for the die designer. POLYM. ENG. SCI., 2009. © 2009 Society of Plastics Engineers     

Influence of slip on the flow of a yield stress fluid around a flat plate

This experimental study deals with the influence of slip on the non‐inertial flow of a viscoplastic fluid around a flat plate moving at a constant velocity. The bulk and interfacial properties of the fluid have been finely characterized. The drag force has been analyzed with regards to the flow velocity and for two tribological conditions: adherence and slip. This force decreases with the velocity and is reduced in the presence of slip. Kinematic fields have also been measured by Particle Image Velocimetry (PIV), to determine the influence of both the velocity and the tribological conditions on the liquid and solid regions of the flow. The results highlight no significant influence of the flow velocity on the thickness of the boundary layer and rigid zones. The wall shear stresses along the plate obtained from force measurements and slip velocities are then compared to rheometrical measurements. © 2015 American Institute of Chemical Engineers AIChE J, 62: 1356–1363, 2016     

Wall slip heating

When molten plastic is extruded, the upper limiting throughput is often dictated by fine irregular distortions of the extrudate surface. Called sharkskin melt fracture, plastics engineers spike plastics formulations with processing aids to suppress these distortions. Sharkskin melt fracture is not to be confused with gross melt fracture, a larger scale distortion arising at throughputs higher than the critical throughput for sharkskin melt fracture. Sharkskin melt fracture has been attributed to a breakdown of the no slip boundary condition in the extrusion die, that is, adhesive failure at the die walls, where the fluid moves with respect to the wall. In this article, we account for the frictional heating at the wall, which we call slip heating. We focus on slit flow, which is used in film casting, sheet extrusion, curtain coating, and when curvature can be neglected, slit flow is easily extended to pipe extrusion and film blowing. In slit flow, the magnitude of the heat flux from the slipping interface is the product of the shear stress and the slip speed. We present the solutions for the temperature rise in pressure‐driven slit flow and simple shearing flow, each subject to constant heat generation at the adhesive slip interface, with and without viscous dissipation in the bulk fluid. We solve the energy equation in Cartesian coordinates for the temperature rise, for steady temperature profiles. For this simplest relevant nonisothermal model, we neglect convective heat transfer in the melt and use a constant viscosity. We arrive at a necessary dimensionless condition for the accurate use of our results: Pé?1. We find that slip heating can raise the melt temperature significantly, as can viscous dissipation in the bulk. We conclude with two worked examples showing the relevance of slip heating in determining wall temperature rise, and we show how to correct wall slip data for this temperature rise. POLYM. ENG. SCI., 55:2042–2049, 2015. © 2014 Society of Plastics Engineers     

Melt fracture of two broad molecular weight distribution high‐density polyethylenes

The melt fracture instabilities of two broad molecular weight distribution (MWD) high‐density polyethylenes (one Ziegler–Natta and one metallocene HDPEs) are studied as functions of the temperature and geometrical details and type of die (cylindrical, slit, and annular). It is found that sharkskin and other melt fracture phenomena are distinctly different for these resins, despite their almost identical rheology. It is also found that the critical conditions for the onset of various melt fracture phenomena depend significantly on the type of die used for their study. For example, sharkskin melt fracture in slit and capillary extrusion was obtained at much small critical shear stress values compared with those found in annular extrusion. Moreover, the metallocene HDPE shows significant slip at the die wall in the sharkskin flow regime. On the other hand, the Ziegler–Natta HDPE has shown no sign of slip. These differences are discussed on the basis of differences in their MWDs that influence their melt elasticity. POLYM. ENG. SCI., 2011. © 2011 Society of Plastics Engineers     

Steady bi‐directional drag flow of non‐newtonian liquids in shallow channels

This paper reexamines the analytical framework for the steady, fully developed two‐dimensional flow in shallow channels, typical of screw extruders, and provides comprehensive solution sets for the Carreau‐Yasuda and the Ostwald‐deWaele model fluids. An application example for polymethylmethacrylate (PMMA) is included to compare the relative merits of the two models under specific extrusion conditions. The Ostwald‐deWaele model, standard in engineering practice, can produce gross misrepresentations of the flow in extruder channels. Particularly, when dealing with highly shear‐sensitive polymers, n < 0.3, or polymers with extended Newtonian plateau, ∧ ～ 1, the Carreau‐Yasuda model becomes preferable. The inclusion of the Carreau model does not necessarily add to the complexity of the analysis.     

Capillary and slit flow behavior of concentrated solutions of a rod-like polymer

This paper is concerned with the flow behavior of isotropic solutions of the rod-like polymer, poly(p-phenyleneterephthalamide) (PPT), in 100 percent sulfuric acid. Studies include entry flow visualization in a slit die and solution fracture, and die swell in capillaries and a slit die. It was observed that solutions of PPT exhibit nearly negligible die swell, a slip-stick type of fracture that disappears at high shear rates, and radial entry flow patterns similar to Newtonian fluids. Fracture was associated with the plateau in the shear stress vs shear rate curve. Because values of the wall shear stress (τ_w.) obtained from capillary measurements were in good agreement with those obtained from a cone-and-plate rheometer and values of the loss modulus (G″) obtained from small-strain dynamic oscillatory measurements, it is believed that the rheological properties lead to the flow instability. These results are in agreement with the predictions of a recent theory by Doi and Edwards for concentrated solutions of rod-like molecules. Data are also presented for a flexible chain polyamide, nylon 6,6, in 100 percent H₂SO₄ for the purpose of comparing the flow characteristics of rigid and flexible chain polymers.     

Velocity measurements on a polypropylene melt during extrusion through a flat coat‐hanger die

An experimental investigation of various flow regimes observed during the extrusion of a polypropylene melt through a flat coat‐hanger die by laser‐Doppler velocimetry (LDV) is presented. LDV measurements of the velocity profiles across the gap of the die at various locations along the die reveal three different extrusion regimes. At small wall shear stresses, the velocity profiles can be fitted by symmetrical curves with the velocities becoming zero at the die walls. These profiles are not uniformly distributed along the die. An increase of the wall shear stress reveals a second flow regime characterized by a uniform distribution of the velocity profiles along the die. As the wall shear stress is increased even further, a third flow regime characterized by wall slip on the glass windows is observed. This flow regime is systematically characterized by measurements of the slip velocities at various temperatures and throughputs. The maximum velocities along the die are taken to assess the uniformity of flow which decisively influences the thickness of the extruded film. By measuring velocity profiles, at different throughput, and temperatures, the conditions for constant velocities along the die were determined. POLYM. ENG. SCI., 2012. © 2011 Society of Plastics Engineers     

Studies on multilayer film coextrusion I. The rheology of flat film coextrusion

Multilayer flat film coextrusion was studied, both experimentally and theoretically. For the experimental study, a sheet-forming die with a feedblock was designed, and plastic films of three and five layers were coextruded. The die was provided with three pressure transducers in the axial direction in order to determine the pressure gradient in the die, allowing the determination of the reduction in pressure drop when different combinations of two polymer melts were coextruded. Polymers used for coextrusion were: (1) low density polyethylene and ethylene-vinyl acetate; (2) low density-polyethylene and high density polyethylene; (3) low density polyethylene and polystyrene. For the theoretical study, the z-component of the equations of motion for steady fully-developed flow were solved using a power law non-Newtonian model, Comparisons were made between the experimental and the theoretically predicted volumetric flow rates. Predictions of the velocity distributions, shear rate profiles, and shear stress distributions were made as functions of the processing conditions and the rheological properties of the individual polymers concerned.     

Experiments and flow analysis of a micropelletizing die

A study has been performed to examine the process of micropelletization on four different polyethylenes with melt index values between 1 and 5 g/10 min. The experiments were done on a 50‐mm 30:1 L/D extruder with an underwater micropelletizer attached. The average micropellet size that was produced ranged from 0.4 to 0.5 mm in diameter depending on whether a plastomer or high‐density grade was selected. The dimensions of the pellets were influenced strongly by the occurrence of die‐hole freeze‐off. Minor sharkskin was observed on the surface of the micropellets, a result of the high stresses experienced in the pelletizer die. A non‐isothermal, axisymmetric flow model was used to assist in the analysis by comparing the observed results to the predicted shear stresses in the die. The calculations revealed that extremely high shear rates were present in the die holes, resulting in a significant degree of wall slip. The measured rheological properties of the micropellets did not show any change in comparison to their virgin resins, likely because of the presence of wall slippage and the short residence time of the polymer in the die holes. Polym. Eng. Sci. 44:1391–1402, 2004. © 2004 Society of Plastics Engineers.     

Melt strength,local velocity,and elongational viscosity profiles of low‐density polyethylene filaments affected by the die design and process conditions

An experimental arrangement to simultaneously measure the melt strength, velocity profiles, and elongational viscosity profiles across the cross section of a molten filament that emerged from either a circular or slit die for low‐density polyethylene (LDPE) under nonisothermal and isothermal conditions is proposed. The proposed experimental rig was based on a parallel coextrusion technique of colored LDPE melt layers into an uncolored melt flowing from the barrel into and out of a die to form a continuous filament before they were pulled down by mechanical rollers until the filament failed. The experimental rig was also equipped with a high‐speed data‐logging system and a personal computer for real‐time measurements. The results suggest that the draw‐down forces changed continuously with changing roller speed, and the velocity profiles of the melt were not uniform across the LDPE filament during the stretching of the melt. Greater draw‐down forces and local melt velocities were obtained in the slit die or under the nonisothermal condition. The draw‐down forces and velocity profiles in both dies were affected by the volumetric flow rates from the extruder and the roller speeds used, with the effect being more pronounced for the circular die. The elongational viscosity profiles of the LDPE filament were not uniform across the filament cross section and corresponded well to the obtained velocity profiles. The elongational viscosities of the LDPE filament were relatively higher when the filament was extruded and stretched in the circular die and under the nonisothermal condition. The changes in the elongational viscosity profiles were more sensitive to changes in the volumetric flow rate and roller speed in the circular die. © 2011 Wiley Periodicals, Inc. J Appl Polym Sci, 2012